I have recently been trying to get into some games which are more demanding than my current laptop can handle. I don't know a whole lot about the technical side of things, for example if everything is going to be compatible with everything else, or what case to get, etc.
The parts I intend to purchase are as follows:

I also have a HDMI monitor, USB keyboard, USB microphone, USB mouse, and headset in mind but they are more generic so I feel as though they are less necessary to mention specifically.

This all brings me to about 600-700 GBP. I am not sure if all parts are going to be compatible (or a good idea to use, for example is one part considerably out of place) with eachother, what case to get, and if there is anything else (other than a HDMI cable) I am going to need. Don't assume I know the obvious, either.
One final note is that I do intend on overclocking some time in the future and I have tried to choose parts which reflect that, but again, I am not certain.

Titan

The PSU is too small for the system. The SAPPHIRE PULSE ITX RX 570 requires at least a 450W. And if you are adding a 200W TDP after-market cooler, you must plan on OC'ing. That would increase the PSU size a bit more. Maybe 500-550W min.

The PSU is too small for the system. The SAPPHIRE PULSE ITX RX 570 requires at least a 450W. And if you are adding a 200W TDP after-market cooler, you must plan on OC'ing. That would increase the PSU size a bit more. Maybe 500-550W min.

So if I use a 500W PSU everything should work well? Also how do I choose a PSU? There's loads of different brands and I'm not sure which to go with.

Titan

The PSU is too small for the system. The SAPPHIRE PULSE ITX RX 570 requires at least a 450W. And if you are adding a 200W TDP after-market cooler, you must plan on OC'ing. That would increase the PSU size a bit more. Maybe 500-550W min.

So if I use a 500W PSU everything should work well? Also how do I choose a PSU? There's loads of different brands and I'm not sure which to go with.

If you're planning on OC'ing the CPU and/or the GPU, 500W would be the minimum, yes. Always look for a strong +12v rail when buying a PSU. Good names are Seasonic, Evga, XFX, Corsair to name a few. You'll probably need to spend at least $50+ to get a good enough one for OC'ing. This Seasonic is a low cost, solid performer.